Extinction and quasi-stationarity in the Verhulst logistic model

A paper with the above title was published in the Journal of Theoretical Biology, Volume 211, pg 11-27, in July, 2001. This paper describes various results for the stochastic version of the Verhulst logistic model, but mathematical proofs and derivations are essentially absent. An expanded version of the paper, including proofs and derivations, is found in pdf-form right here . This web-manuscript is illustrated by eight figures. Matlab programs have been used for the numerical evaluations in these figures. If you wish to see these Matlab programs, click here.

A study of the moments of the quasi-stationary distribution of the same model was undertaken in 2002. The study is confined to the parameter region where R0 is distinctly larger than one. The results are described in the paper "Moment closure and the stochastic logistic model", which appeared in Theoretical Population Biology, Vol 63, 159-168, 2003. The mathematical development is briefly described in an appendix of this paper. I have used Maple to derive the asymptotic approximations of the first three cumulants. The Maple worksheet that shows these derivations is found here.

The so-called logistic SIS model is a special case of the Verhulst model. A detailed study of its quasi-stationary distribution and of the accompanying time to extinction is given in a monograph with the title "Extinction and Quasi-stationarity in the Stochastic Logistic SIS Model", Springer Lecture Notes in Mathematics #2022, 2011, by Ingemar Nåsell. In several respects the results in this monograph are improvements over the corresponding results for the Verhulst model. The interested reader is recommended to consult the quoted monograph. Comments and corrections on this monograph are found here.

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